"Anarchism is a tendency in the history of human thought & action which seeks to identify coercive, authoritarian, & hierarchic structures of all kinds & to challenge their legitimacy — & if they cannot justify their legitimacy, which is quite commonly the case, to work to undermine them & expand the scope of freedom." — Noam Chomsky
"At some point we must draw a line across the ground of our home & our being, drive a spear into the land, & say to the bulldozers, earthmovers, government & corporations, "thus far & no farther." — Edward Abbey
Anarchism is a political belief system that supports decentralized activity and governance, against centralized organization and hierarchy. Another way of putting it is that anarchism is opposed to the State and capitalism, and supports grassroots social ownership of a society's resources.
This may take many forms: employee-owned and managed cooperatives; community councils; agricultural communes with shared equipment; consensus-based affinity groups. The key is simply direct power over the mechanisms that control your life, mechanisms of work, government, child-rearing, art and culture, whatever. This control can only be exercised socially, assuming that production and governance is social. Instead of individualism or collectivism, anarchism champions democratic community-self-organization.
The history of modern anarchism spans 150 years, and is represented most dramatically in events such as the Spanish Revolution of 1936 or the London Poll Tax revolt of the 70s. Yet it is a singular and unusual politics that escapes the normal boundaries of political consideration, and seeks application across the full range of human experience. Anywhere that domination is possible, there you may see an anarchist current moving against it.